Bifurcation of pressurized functionally graded elastomeric hollow cylinders

Abstract

Various bifurcation behaviors, including asymmetric axial buckling induced by end thrust, bifurcation under external pressure, and eversion without any loading, of a soft circular hollow cylinder (or cylindrical shell) composed of functionally graded incompressible elastomeric material are considered in a unified way. Based on the nonlinear elasticity theory of a deformable continuous body undergoing large deformation and the linearized incremental theory for a superimposed infinitesimal deformation, an efficient approach for buckling analysis is developed and applied to the cylinder subjected to a combination of axial end thrust and internal/external pressure. It is based on the state-space formalism, which naturally avoids the derivatives of instantaneous material constants, enabling a convenient and accurate numerical implementation. Along with the layerwise method, an analytical characteristic equation (i.e. the bifurcation criterion) governing the general asymmetric buckling of the cylinder is derived. Detailed parametric studies are then carried out for a pressurized soft functionally graded hollow cylinder using an incompressible Mooney-Rivlin material model. The effects of material gradient on bifurcation induced either by end thrust or external pressure are discussed through numerical examples. Not only does this study provide an efficient tool for buckling analysis of soft cylinders, some findings that are unique to a functionally graded material are also highlighted. All in all, it is found highly possible to tune the bifurcation behavior of soft elastomeric cylindrical shells by tailoring material composition and/or adjusting the pressures acting on the surfaces of the cylinder.

Fate of a bulge in an inflated hyperelastic tube: theory and experiment

“…The incompressibility condition is enforced through a Lagrange multiplier. To conduct the buckling analysis, we follow [40,41]. First, we introduce the incremental state vector U = [ u r , u θ , u z , Σ rr , Σ rθ , Σ rz ] T .…”

On propagation of axisymmetric waves in pressurized functionally graded elastomeric hollow cylinders

“…In the middle of 1980s, a new type of materials called functionally graded materials (FGMs), which exhibit inhomogeneous material properties varying continuously along one or more directions, was first proposed and prepared by a group of Japanese material scientists. FGMs have been designed for various purposes of reducing the residual and thermal stresses, increasing the bonding strength, eliminating the sharp stress discontinuity, and improving the efficiency and life of acoustic wave devices [31,32]. Because of the excellent performance and superiority over many other conventional materials, FGMs have gained widespread applications not only in aerospace engineering but also in biology [33,34].…”

Electrostatically tunable axisymmetric vibrations of soft electro-active tubes

“…Consequently, the resulting incremental governing equations are a system of coupled partial differential equations with variable coefficients, which are difficult to solve analytically or even numerically via the conventional displacement-based method. Therefore, the state-space method (SSM) [46,52,53] combining the state-space formalism with the approximate laminate technique is adopted in this paper to derive the frequency equations of the axisymmetric vibrations of SEA tubes.…”

“…, the global stiffness reduces rapidly so that barrelling instabilities [52,55] occur in the SEA tube. This is why the frequencies of modes 1 n  and 2 n  in Figures 7(b-c) decrease gently at first and then dramatically.…”